Abstract

The dispersion equation for surface waves—with simple transverse exponential decay at the interface of identical biaxial crystals with a relative twist about the axis normal to the interface and propagating along a bisector of the angle between the crystallographic configurations on either side of the interface—has several solutions of which only one is physical. The selected type of surface wave is possible only for a restricted range of the twist angle, which depends on the ratio of the maximum and the minimum of the principal refractive indexes and the angle between the optic ray axes.

© 2007 Optical Society of America

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References

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  1. A. D. Boardman, ed., Electromagnetic Surface Modes (Wiley, 1982).
  2. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).
  3. L. Torner, J. P. Torres, and D. Mihalache, "New type of guided waves in birefringent media," IEEE Photon. Technol. Lett. 5, 201-203 (1993).
    [CrossRef]
  4. J. Pendry, "Applied physics: enhanced: playing tricks with light," Science 10, 1687-1688 (1999).
    [CrossRef]
  5. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
    [CrossRef] [PubMed]
  6. W.S. Weiglhofer and A. Lakhtakia, eds., Introduction to Complex Mediums for Optics and Electromagnetics (SPIE, 2003).
    [CrossRef]
  7. M. I. D'yakonov, "New type of electromagnetic wave propagating at an interface," Sov. Phys. JETP 67, 714-716 (1988).
  8. N. S. Averkiev and M. I. Dyakonov, "Electromagnetic waves localized at the interface of transparent anisotropic media," Opt. Spectrosc. 68, 653-655 (1990).
  9. D. B. Walker, E. N. Glytsis, and T. K. Gaylord, "Surface mode at isotropic-uniaxial and isotropic-biaxial interfaces," J. Opt. Soc. Am. A 15, 248-260 (1998).
    [CrossRef]
  10. A. N. Furs, V. M. Galynsky, and L. M. Barkovsky, "Surface polaritons in symmetry planes of biaxial crystals," J. Phys. A 38, 8083-8101 (2005).
    [CrossRef]
  11. A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "The relation between crystallographic symmetry and the bidirectionality of SAW generation by IDT's," in Proceedings of the IEEE Ultrasonics Symposium (IEEE, 2002), pp. 223-226.
  12. A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "Fundamental frequency degeneracy of standing surface acoustic waves under metallic gratings on piezoelectric substrates," J. Acoust. Soc. Am. 112, 2003-2013 (2002).
    [CrossRef] [PubMed]
  13. A. N. Darinskiĭ, "Dispersionless polaritons on a twist boundary in optically uniaxial crystals," Crystallogr. Rep. 46, 842-844 (2001).
    [CrossRef]
  14. C. D. Gribble and A. J. Hall, Optical Mineralogy: Principles and Practice (UCL, 1993).
  15. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
    [CrossRef]
  16. M. Berry, R. Bhandari, and S. Klein, "Black plastic sandwiches demonstrating biaxial optical anisotropy," Eur. J. Phys. 20, 1-14 (1999).
    [CrossRef]
  17. B. Michel, "Recent developments in the homogenization of linear bianisotropic composite materials," in Electromagnetic Fields in Unconventional Materials and Structures, O.N.Singh and A.Lakhtakia, eds. (Wiley, 2000), pp. 39-82.
  18. T. G. Mackay and W. S. Weiglhofer, "Homogenization of biaxial composite materials: bianisotropic properties," J. Opt. A, Pure Appl. Opt. 3, 45-52 (2001).
    [CrossRef]
  19. W. S. Weiglhofer, "Constitutive characterization of simple and complex mediums," in Introduction to Complex Mediums for Optics and Electromagnetics, W.S.Weiglhofer and A.Lakhtakia, eds. (SPIE, 2003), pp. 27-61.
    [CrossRef]
  20. W. S. Weiglhofer and A. Lakhtakia, "On electromagnetic waves in biaxial bianisotropic media," Electromagnetics 19, 351-362 (1999).
    [CrossRef]
  21. If we write E=r-1=aI=+b(m̂1m̂2+m̂2m̂1), then the unit vectors m̂1 and m̂2 are parallel to the two optic axes of a biaxial dielectric material.
  22. Mineralogy Database, http://www.webmineral.com/.
  23. R. F. Wallis, "Optical properties associated with surface excitations of semiconductors," in Optical Properties of Semiconductors, M.Balkanski, ed. (Elsevier North-Holland, 1994), Vol. 2, pp. 1-31.
  24. There appears to be a typographical error in Darinskiĭ's publication [Ref. , Eq. (10)]. To make it consistent with our results and the author's own Fig. , it should read V2/Ve2=E]-[1−3sin2phiv+cos2phiv[1+4sin2phiv(E-−1)]½]/{2[1+(E]-−1)sin]2phiv][E]-cos]4phiv−sin]4phiv]}.
  25. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1981).
  26. C. Klein and C. S. Hurlbut, Jr., Manual of Mineralogy, 20th ed. (Wiley, 1977), pp. 249-250.

2005

A. N. Furs, V. M. Galynsky, and L. M. Barkovsky, "Surface polaritons in symmetry planes of biaxial crystals," J. Phys. A 38, 8083-8101 (2005).
[CrossRef]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

2002

A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "Fundamental frequency degeneracy of standing surface acoustic waves under metallic gratings on piezoelectric substrates," J. Acoust. Soc. Am. 112, 2003-2013 (2002).
[CrossRef] [PubMed]

2001

A. N. Darinskiĭ, "Dispersionless polaritons on a twist boundary in optically uniaxial crystals," Crystallogr. Rep. 46, 842-844 (2001).
[CrossRef]

T. G. Mackay and W. S. Weiglhofer, "Homogenization of biaxial composite materials: bianisotropic properties," J. Opt. A, Pure Appl. Opt. 3, 45-52 (2001).
[CrossRef]

1999

W. S. Weiglhofer and A. Lakhtakia, "On electromagnetic waves in biaxial bianisotropic media," Electromagnetics 19, 351-362 (1999).
[CrossRef]

M. Berry, R. Bhandari, and S. Klein, "Black plastic sandwiches demonstrating biaxial optical anisotropy," Eur. J. Phys. 20, 1-14 (1999).
[CrossRef]

J. Pendry, "Applied physics: enhanced: playing tricks with light," Science 10, 1687-1688 (1999).
[CrossRef]

1998

1993

L. Torner, J. P. Torres, and D. Mihalache, "New type of guided waves in birefringent media," IEEE Photon. Technol. Lett. 5, 201-203 (1993).
[CrossRef]

1990

N. S. Averkiev and M. I. Dyakonov, "Electromagnetic waves localized at the interface of transparent anisotropic media," Opt. Spectrosc. 68, 653-655 (1990).

1988

M. I. D'yakonov, "New type of electromagnetic wave propagating at an interface," Sov. Phys. JETP 67, 714-716 (1988).

Averkiev, N. S.

N. S. Averkiev and M. I. Dyakonov, "Electromagnetic waves localized at the interface of transparent anisotropic media," Opt. Spectrosc. 68, 653-655 (1990).

Barkovsky, L. M.

A. N. Furs, V. M. Galynsky, and L. M. Barkovsky, "Surface polaritons in symmetry planes of biaxial crystals," J. Phys. A 38, 8083-8101 (2005).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Berry, M.

M. Berry, R. Bhandari, and S. Klein, "Black plastic sandwiches demonstrating biaxial optical anisotropy," Eur. J. Phys. 20, 1-14 (1999).
[CrossRef]

Bhandari, R.

M. Berry, R. Bhandari, and S. Klein, "Black plastic sandwiches demonstrating biaxial optical anisotropy," Eur. J. Phys. 20, 1-14 (1999).
[CrossRef]

Biryukov, S. V.

A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "Fundamental frequency degeneracy of standing surface acoustic waves under metallic gratings on piezoelectric substrates," J. Acoust. Soc. Am. 112, 2003-2013 (2002).
[CrossRef] [PubMed]

A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "The relation between crystallographic symmetry and the bidirectionality of SAW generation by IDT's," in Proceedings of the IEEE Ultrasonics Symposium (IEEE, 2002), pp. 223-226.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1981).

Darinskii, A. N.

A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "Fundamental frequency degeneracy of standing surface acoustic waves under metallic gratings on piezoelectric substrates," J. Acoust. Soc. Am. 112, 2003-2013 (2002).
[CrossRef] [PubMed]

A. N. Darinskiĭ, "Dispersionless polaritons on a twist boundary in optically uniaxial crystals," Crystallogr. Rep. 46, 842-844 (2001).
[CrossRef]

A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "The relation between crystallographic symmetry and the bidirectionality of SAW generation by IDT's," in Proceedings of the IEEE Ultrasonics Symposium (IEEE, 2002), pp. 223-226.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Dyakonov, M. I.

N. S. Averkiev and M. I. Dyakonov, "Electromagnetic waves localized at the interface of transparent anisotropic media," Opt. Spectrosc. 68, 653-655 (1990).

D'yakonov, M. I.

M. I. D'yakonov, "New type of electromagnetic wave propagating at an interface," Sov. Phys. JETP 67, 714-716 (1988).

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Furs, A. N.

A. N. Furs, V. M. Galynsky, and L. M. Barkovsky, "Surface polaritons in symmetry planes of biaxial crystals," J. Phys. A 38, 8083-8101 (2005).
[CrossRef]

Galynsky, V. M.

A. N. Furs, V. M. Galynsky, and L. M. Barkovsky, "Surface polaritons in symmetry planes of biaxial crystals," J. Phys. A 38, 8083-8101 (2005).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Gribble, C. D.

C. D. Gribble and A. J. Hall, Optical Mineralogy: Principles and Practice (UCL, 1993).

Hall, A. J.

C. D. Gribble and A. J. Hall, Optical Mineralogy: Principles and Practice (UCL, 1993).

Hurlbut, C. S.

C. Klein and C. S. Hurlbut, Jr., Manual of Mineralogy, 20th ed. (Wiley, 1977), pp. 249-250.

Klein, C.

C. Klein and C. S. Hurlbut, Jr., Manual of Mineralogy, 20th ed. (Wiley, 1977), pp. 249-250.

Klein, S.

M. Berry, R. Bhandari, and S. Klein, "Black plastic sandwiches demonstrating biaxial optical anisotropy," Eur. J. Phys. 20, 1-14 (1999).
[CrossRef]

Lakhtakia, A.

W. S. Weiglhofer and A. Lakhtakia, "On electromagnetic waves in biaxial bianisotropic media," Electromagnetics 19, 351-362 (1999).
[CrossRef]

Mackay, T. G.

T. G. Mackay and W. S. Weiglhofer, "Homogenization of biaxial composite materials: bianisotropic properties," J. Opt. A, Pure Appl. Opt. 3, 45-52 (2001).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

Michel, B.

B. Michel, "Recent developments in the homogenization of linear bianisotropic composite materials," in Electromagnetic Fields in Unconventional Materials and Structures, O.N.Singh and A.Lakhtakia, eds. (Wiley, 2000), pp. 39-82.

Mihalache, D.

L. Torner, J. P. Torres, and D. Mihalache, "New type of guided waves in birefringent media," IEEE Photon. Technol. Lett. 5, 201-203 (1993).
[CrossRef]

Pendry, J.

J. Pendry, "Applied physics: enhanced: playing tricks with light," Science 10, 1687-1688 (1999).
[CrossRef]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

Torner, L.

L. Torner, J. P. Torres, and D. Mihalache, "New type of guided waves in birefringent media," IEEE Photon. Technol. Lett. 5, 201-203 (1993).
[CrossRef]

Torres, J. P.

L. Torner, J. P. Torres, and D. Mihalache, "New type of guided waves in birefringent media," IEEE Photon. Technol. Lett. 5, 201-203 (1993).
[CrossRef]

Walker, D. B.

Wallis, R. F.

R. F. Wallis, "Optical properties associated with surface excitations of semiconductors," in Optical Properties of Semiconductors, M.Balkanski, ed. (Elsevier North-Holland, 1994), Vol. 2, pp. 1-31.

Weiglhofer, W. S.

T. G. Mackay and W. S. Weiglhofer, "Homogenization of biaxial composite materials: bianisotropic properties," J. Opt. A, Pure Appl. Opt. 3, 45-52 (2001).
[CrossRef]

W. S. Weiglhofer and A. Lakhtakia, "On electromagnetic waves in biaxial bianisotropic media," Electromagnetics 19, 351-362 (1999).
[CrossRef]

W. S. Weiglhofer, "Constitutive characterization of simple and complex mediums," in Introduction to Complex Mediums for Optics and Electromagnetics, W.S.Weiglhofer and A.Lakhtakia, eds. (SPIE, 2003), pp. 27-61.
[CrossRef]

Weihnacht, M.

A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "Fundamental frequency degeneracy of standing surface acoustic waves under metallic gratings on piezoelectric substrates," J. Acoust. Soc. Am. 112, 2003-2013 (2002).
[CrossRef] [PubMed]

A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "The relation between crystallographic symmetry and the bidirectionality of SAW generation by IDT's," in Proceedings of the IEEE Ultrasonics Symposium (IEEE, 2002), pp. 223-226.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1981).

Crystallogr. Rep.

A. N. Darinskiĭ, "Dispersionless polaritons on a twist boundary in optically uniaxial crystals," Crystallogr. Rep. 46, 842-844 (2001).
[CrossRef]

Electromagnetics

W. S. Weiglhofer and A. Lakhtakia, "On electromagnetic waves in biaxial bianisotropic media," Electromagnetics 19, 351-362 (1999).
[CrossRef]

Eur. J. Phys.

M. Berry, R. Bhandari, and S. Klein, "Black plastic sandwiches demonstrating biaxial optical anisotropy," Eur. J. Phys. 20, 1-14 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

L. Torner, J. P. Torres, and D. Mihalache, "New type of guided waves in birefringent media," IEEE Photon. Technol. Lett. 5, 201-203 (1993).
[CrossRef]

J. Acoust. Soc. Am.

A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "Fundamental frequency degeneracy of standing surface acoustic waves under metallic gratings on piezoelectric substrates," J. Acoust. Soc. Am. 112, 2003-2013 (2002).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt.

T. G. Mackay and W. S. Weiglhofer, "Homogenization of biaxial composite materials: bianisotropic properties," J. Opt. A, Pure Appl. Opt. 3, 45-52 (2001).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

A. N. Furs, V. M. Galynsky, and L. M. Barkovsky, "Surface polaritons in symmetry planes of biaxial crystals," J. Phys. A 38, 8083-8101 (2005).
[CrossRef]

Nature

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Opt. Spectrosc.

N. S. Averkiev and M. I. Dyakonov, "Electromagnetic waves localized at the interface of transparent anisotropic media," Opt. Spectrosc. 68, 653-655 (1990).

Science

J. Pendry, "Applied physics: enhanced: playing tricks with light," Science 10, 1687-1688 (1999).
[CrossRef]

Sov. Phys. JETP

M. I. D'yakonov, "New type of electromagnetic wave propagating at an interface," Sov. Phys. JETP 67, 714-716 (1988).

Other

W.S. Weiglhofer and A. Lakhtakia, eds., Introduction to Complex Mediums for Optics and Electromagnetics (SPIE, 2003).
[CrossRef]

A. D. Boardman, ed., Electromagnetic Surface Modes (Wiley, 1982).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

A. N. Darinskiĭ, S. V. Biryukov, and M. Weihnacht, "The relation between crystallographic symmetry and the bidirectionality of SAW generation by IDT's," in Proceedings of the IEEE Ultrasonics Symposium (IEEE, 2002), pp. 223-226.

C. D. Gribble and A. J. Hall, Optical Mineralogy: Principles and Practice (UCL, 1993).

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

W. S. Weiglhofer, "Constitutive characterization of simple and complex mediums," in Introduction to Complex Mediums for Optics and Electromagnetics, W.S.Weiglhofer and A.Lakhtakia, eds. (SPIE, 2003), pp. 27-61.
[CrossRef]

B. Michel, "Recent developments in the homogenization of linear bianisotropic composite materials," in Electromagnetic Fields in Unconventional Materials and Structures, O.N.Singh and A.Lakhtakia, eds. (Wiley, 2000), pp. 39-82.

If we write E=r-1=aI=+b(m̂1m̂2+m̂2m̂1), then the unit vectors m̂1 and m̂2 are parallel to the two optic axes of a biaxial dielectric material.

Mineralogy Database, http://www.webmineral.com/.

R. F. Wallis, "Optical properties associated with surface excitations of semiconductors," in Optical Properties of Semiconductors, M.Balkanski, ed. (Elsevier North-Holland, 1994), Vol. 2, pp. 1-31.

There appears to be a typographical error in Darinskiĭ's publication [Ref. , Eq. (10)]. To make it consistent with our results and the author's own Fig. , it should read V2/Ve2=E]-[1−3sin2phiv+cos2phiv[1+4sin2phiv(E-−1)]½]/{2[1+(E]-−1)sin]2phiv][E]-cos]4phiv−sin]4phiv]}.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1981).

C. Klein and C. S. Hurlbut, Jr., Manual of Mineralogy, 20th ed. (Wiley, 1977), pp. 249-250.

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Figures (13)

Fig. 1
Fig. 1

Geometry of the biaxial bicrystalline interface. The optic ray axes in region A ( z > 0 ) are parallel to the unit vectors c ̂ 1 A and c ̂ 2 A , whereas the corresponding unit vectors in region B ( z < 0 ) are c ̂ 1 B and c ̂ 2 B .

Fig. 2
Fig. 2

v ¯ as a function of ξ for solution 3B when ϵ ¯ = 1.6 , and δ = 0 ° , 20 ° , and 40 ° .

Fig. 3
Fig. 3

¯ 1 2 as a function of ξ for solution 3B when ϵ ¯ = 1.6 , and δ = 0 ° , 20 ° , and 40 ° .

Fig. 4
Fig. 4

¯ 2 2 as a function of ξ for solution 3B when ϵ ¯ = 1.6 , and δ = 0 ° , 20 ° , and 40 ° .

Fig. 5
Fig. 5

ϵ max as a function of ϵ ¯ for solution 3B when δ = 0.43 ° , 10 ° , 20 ° , 40 ° , and 60 ° .

Fig. 6
Fig. 6

v ¯ as a function of ξ for solution 3B when ϵ ¯ = 5 , and δ = 2 ° .

Fig. 7
Fig. 7

¯ 1 2 as a function of ξ for solution 3B when ϵ ¯ = 5 , and δ = 2 ° .

Fig. 8
Fig. 8

Zones of surface-wave propagation with real-valued transverse decay constant (white area) in the ϵ ¯ ξ plane: (a) δ = 0.5 ° , and (b) δ = 2 ° .

Fig. 9
Fig. 9

Hemimorphite: (a) 1 v ¯ and (b) p ¯ 1 2 and p ¯ 2 2 as functions of ξ.

Fig. 10
Fig. 10

Crocoite: (a) 1 v ¯ and (b) p ¯ 1 2 and p ¯ 2 2 as functions of ξ.

Fig. 11
Fig. 11

Tellurite: (a) 1 v ¯ and (b) p ¯ 1 2 and p ¯ 2 2 as functions of ξ.

Fig. 12
Fig. 12

Witherite:(a) 1 v ¯ and (b) p ¯ 1 2 and p ¯ 2 2 as functions of ξ.

Fig. 13
Fig. 13

Cerussite: (a) 1 v ¯ and (b) p ¯ 1 2 and p ¯ 2 2 as functions of ξ.

Equations (49)

Equations on this page are rendered with MathJax. Learn more.

ϵ ͇ r = α I ͇ + β ( c 1 ̂ c 2 ̂ + c 2 ̂ c 1 ̂ ) ,
α = n b 2
β = n c 2 n a 2 2 .
δ = cos 1 [ n c 2 n b 2 n c 2 n a 2 ] .
ϵ ͇ r A = [ α + β [ cos ( 2 δ ) cos ( 2 ξ ) ] β sin ( 2 ξ ) 0 β sin ( 2 ξ ) α + β [ cos ( 2 δ ) + cos ( 2 ξ ) ] 0 0 0 α ]
ϵ ͇ r B = [ α + β [ cos ( 2 δ ) cos ( 2 ξ ) ] β sin ( 2 ξ ) 0 β sin ( 2 ξ ) α + β [ cos ( 2 δ ) + cos ( 2 ξ ) ] 0 0 0 α ] ,
k ̱ = κ x ̂ + i p z ̂ ,
k ̱ × E ̱ = ω μ o H ̱ ,
k ̱ × H ̱ = ω ϵ o ϵ ͇ r E ̱ ,
p 1 2 = A 1 + A 2 κ 2 A 3 + A 4 κ 2 + A 5 κ 4
p 2 2 = A 1 + A 2 κ 2 + A 3 + A 4 κ 2 + A 5 κ 4 ,
A 1 = k o 2 [ α + β cos ( 2 δ ) ] ,
A 2 = 2 α + β [ cos ( 2 δ ) cos ( 2 ξ ) ] 2 α ,
A 3 = k o 4 β 2 ,
A 4 = k o 2 β 2 ( cos ( 2 δ ) cos ( 2 ξ ) 1 α ) ,
A 5 = ( β [ cos ( 2 δ ) cos ( 2 ξ ) ] 2 α ) 2 ,
k o = ω ϵ o μ o
E ̱ A l = A l x ̂ + A l k o 2 β sin ( 2 ξ ) p l 2 + k o 2 α κ 2 + k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] y ̂ + i A l p l κ κ 2 k o 2 α z ̂ , l = 1 , 2 ,
H ̱ A l = i A l k o 2 p l β sin ( 2 ξ ) ω μ o { p l 2 + k o 2 α κ 2 + k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] } x ̂ i A l p l α ϵ o ω κ 2 k o 2 α y ̂ + A l k o 2 β κ sin ( 2 ξ ) ω μ o { p l 2 + k o 2 α κ 2 + k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] } z ̂ , l = 1 , 2 ,
E ̱ B l = B l x ̂ B l k o 2 β sin ( 2 ξ ) p l 2 + k o 2 α κ 2 + k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] y ̂ i B l p l κ κ 2 k o 2 α z ̂ , l = 1 , 2 ,
H ̱ B l = i B l k o 2 p l β sin ( 2 ξ ) ω μ 0 { p l 2 + k o 2 α κ 2 + k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] } x ̂ + i B l p l α ϵ o ω κ 2 k o 2 α y ̂ B l k o 2 β κ sin ( 2 ξ ) ω μ o { p l 2 + k o 2 α κ 2 + k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] } z ̂ , l = 1 , 2 ,
x ̂ E ̱ A 1 + x ̂ E ̱ A 2 = x ̂ E ̱ B 1 + x ̂ E ̱ B 2 ,
y ̂ E ̱ A 1 + y ̂ E ̱ A 2 = y ̂ E ̱ B 1 + y ̂ E ̱ B 2 ,
x ̂ H ̱ A 1 + x ̂ H ̱ A 2 = x ̂ H ̱ B 1 + x ̂ H ̱ B 2 ,
y ̂ H ̱ A 1 + y ̂ H ̱ A 2 = y ̂ H ̱ B 1 + y ̂ H ̱ B 2 .
0 = ( p 1 p 2 ) 2 { p 1 2 + p 1 p 2 + p 2 2 + k o 2 α κ 2 + k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] } × { p 1 p 2 k o 2 α + κ 2 k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] } .
( p 1 p 2 ) 2 = 0 ,
κ 2 p 1 2 p 2 2 k o 2 α k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] = p 1 p 2 ,
κ 2 + k o 2 α + k o 2 β [ cos ( 2 δ ) + cos ( 2 ξ ) ] = p 1 p 2 .
B 1 = B 2 = A 2 = A 1 ;
κ 2 = k o 2 α
κ 2 = k o 2 α 4 [ α + β cos ( 2 δ ) ] cos ( 2 ξ ) β [ 3 + cos ( 4 ξ ) ] 2 [ cos ( 2 ξ ) cos ( 2 δ ) ] [ α + β cos ( 2 δ ) β cos ( 2 ξ ) ] .
κ 2 = k o 2 C 0 12 α cos ( 2 ξ ) C 1 + C 2 cos ( 2 ξ ) + C 3 cos ( 4 ξ ) + C 4 cos ( 6 ξ ) 8 [ cos ( 2 δ ) cos ( 2 ξ ) ]
κ 2 = k o 2 C 0 12 α cos ( 2 ξ ) + C 1 + C 2 cos ( 2 ξ ) + C 3 cos ( 4 ξ ) + C 4 cos ( 6 ξ ) 8 [ cos ( 2 δ ) cos ( 2 ξ ) ] ,
C 0 = 4 α cos ( 2 δ ) 4 β sin 2 ( 2 δ ) ,
C 1 = 16 α 2 + 6 β 2 + 24 α β cos ( 2 δ ) + 8 ( α 2 β 2 ) cos ( 4 δ ) + 8 α β cos ( 6 δ ) + 2 β 2 cos ( 8 δ ) ,
C 2 = 32 α 2 cos ( 2 δ ) + 16 α β cos ( 4 δ ) ,
C 3 = 8 α 2 32 α β cos ( 2 δ ) ,
C 4 = 16 α β .
κ 2 = ( k 0 n a 2 sin ξ ) 2 × [ 1 3 cos ( 2 ξ ) + 2 cos 2 ξ 1 + 2 ( n c n a ) 2 2 [ ( n c n a ) 2 1 ] cos ( 2 ξ ) ] ,
v ± = ω k 0 1 2 [ 1 n a 2 + 1 n c 2 + ( 1 n a 2 1 n c 2 ) cos ( θ 1 ± θ 2 ) ] ,
v + = ω k o n b ,
v = ω k o sin 2 ξ n a 2 + cos 2 ξ n c 2 .
v ¯ = v min { v + , v } ,
p ¯ 1 p 1 k o ,
p ¯ 2 p 2 k o ,
ϵ ¯ = ( n c n a ) 2 .
¯ 1 p ¯ 1 n a ,
¯ 2 p ¯ 2 n a ,

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